起源故事 · Motorola 6σ 与 Mikel Harry
Origin Story · Motorola 6σ and Mikel Harry
1980 年代末,Motorola 推 Six Sigma 时碰到一个尴尬:Cpk 用 min(Z_USL, Z_LSL)/3 只看一侧,
如果上侧 ppm = 1000 / 下侧 ppm = 800,Cpk 只反映上侧那 1000,下侧的 800 完全不见了。
Mikel Harry(6σ 总师)于是定义 Z_bench:先把两侧 ppm 相加得 P_total,再反查标准正态找回综合 Z。
从此 Motorola / GE 内部黑带报告统一用 Z_bench——它把"丢失的另一侧"找回来。Minitab 的 Capability Sixpack 默认输出也是 Z_bench。
In the late 1980s, Motorola hit an awkward corner while launching Six Sigma: Cpk = min(Z_USL, Z_LSL) / 3 reports one side only.
If the upper tail leaks 1000 ppm and the lower tail leaks 800 ppm, Cpk reflects the 1000 alone — the 800 disappears.
Mikel Harry, the chief architect of 6σ, defined Z_bench as the fix: sum both ppm into P_total, then invert the standard normal to recover a composite Z.
From that point on, internal Black Belt reports at Motorola and GE used Z_bench — it brings the "missing other side" back into view. Minitab's Capability Sixpack still defaults to Z_bench.
1 两侧 ppm → 总尾部 → 反查 Z_bench Both-tail ppm → Total tail → Invert to Z_bench
看箭头Follow the arrow左半:原始过程钟形,上下尾部各染红/蓝并标 ppm。右半:标准正态 N(0,1),把 P_total 当作单侧尾部反查回 Z_bench。 Left: the raw process bell with upper / lower tails shaded red and blue, each labelled with its ppm. Right: the standard normal N(0,1), where P_total is treated as a single tail and inverted back into Z_bench.
2 Cpk × 3 vs Z_bench 双柱 Cpk × 3 vs Z_bench — Side-by-Side Bars
灰色 = Cpk × 3 = Z_min 只看短板;紫色 = Z_bench 含双侧诚实总账。 μ 越居中两者差距越小;μ 越偏,Z_bench 永远 ≤ Z_min。 Grey = Cpk × 3 = Z_min sees only the short side; purple = Z_bench carries the honest two-tail total. The more centered μ is, the smaller the gap; the more off-center μ is, the more Z_bench stays ≤ Z_min.
3 现实里的 Z_bench Z_bench in the Real World
Motorola / GE 6σ 报告:黑带级项目 capability 页默认报 Z_bench 而非 Cpk,6σ 标杆 Z_bench ≥ 6(短期)/ ≥ 4.5(长期含 1.5σ 漂移)。
Motorola / GE 6σ reports: Black Belt capability pages default to Z_bench instead of Cpk. The 6σ benchmark is Z_bench ≥ 6 (short-term) or ≥ 4.5 (long-term with the 1.5 σ shift).
半导体良率合成:多步工艺的最终 ppm 由每步 ppm 累加,再反查得整体 Z_bench,用于晶圆厂 yield 看板。
Semiconductor yield rollup: final ppm of a multi-step process is the sum of per-step ppm, then inverted into an overall Z_bench for the fab yield dashboard.
Minitab Capability Sixpack:"Z.Bench" 是默认输出栏目,旁边才是 Z.LSL / Z.USL / Cpk / Ppk。
Minitab Capability Sixpack: "Z.Bench" is the headline output column; Z.LSL / Z.USL / Cpk / Ppk sit beside it as supporting figures.
USP <905> 含量均匀度:药典法规用"双侧合成超规率"衡量批次合格性,本质就是 Z_bench 的应用。
USP <905> content uniformity: the pharmacopeia uses two-tail combined defect rate to judge batch acceptance — this is Z_bench applied to drug manufacturing.
一句话In One Line
Cpk 是"短板诚实",Z_bench 是"全员诚实"。
当一侧严重偏 + 另一侧也有少量超规时,Cpk 漏掉了次侧的不良;Z_bench 把两侧 ppm 相加再反查,永远 ≤ Z_min。
关键直觉:Z 不能相加。Z 是分位点不是面积,必须先 Z → ppm(面积)→ 加和 → ppm 反查 → Z_bench。
所以 Six Sigma 黑带级报告默认用 Z_bench:它是双侧合成后唯一不撒谎的"综合 σ 等级"。 Cpk is "honest about the short side"; Z_bench is "honest about everyone".
When one tail is heavily off-center and the other still has a small leak, Cpk misses that secondary leak; Z_bench sums both ppm and inverts back, so it stays ≤ Z_min, always.
Key intuition: Z values do not add. Z is a quantile, not an area — you must first convert Z → ppm (area), sum, invert the ppm, and only then read off Z_bench.
That's why Black Belt reports default to Z_bench: it is the only composite sigma level that doesn't lie after combining both tails.
当一侧严重偏 + 另一侧也有少量超规时,Cpk 漏掉了次侧的不良;Z_bench 把两侧 ppm 相加再反查,永远 ≤ Z_min。
关键直觉:Z 不能相加。Z 是分位点不是面积,必须先 Z → ppm(面积)→ 加和 → ppm 反查 → Z_bench。
所以 Six Sigma 黑带级报告默认用 Z_bench:它是双侧合成后唯一不撒谎的"综合 σ 等级"。 Cpk is "honest about the short side"; Z_bench is "honest about everyone".
When one tail is heavily off-center and the other still has a small leak, Cpk misses that secondary leak; Z_bench sums both ppm and inverts back, so it stays ≤ Z_min, always.
Key intuition: Z values do not add. Z is a quantile, not an area — you must first convert Z → ppm (area), sum, invert the ppm, and only then read off Z_bench.
That's why Black Belt reports default to Z_bench: it is the only composite sigma level that doesn't lie after combining both tails.
常见误用Common Mistakes
Z_bench = Z_USL + Z_LSL 直接加 Z。错!Z 是分位点不是面积,不能加。必须先各自转 ppm → 加和 → 反查 Φ⁻¹ 得 Z_bench。
Computing Z_bench = Z_USL + Z_LSL by adding Z values. Wrong — Z is a quantile, not an area, and quantiles don't add. Convert each to ppm first, sum, then apply Φ⁻¹ to recover Z_bench.
Z_bench = Cpk × 3 = Z_min。只在严重偏心(一侧 ppm 远大于另一侧)时才近似相等。居中时 Z_bench 略小于 Z_min(少量另一侧 ppm 拉低)。
Assuming Z_bench = Cpk × 3 = Z_min. The two only converge when μ is heavily off-center (one tail dwarfs the other). When μ is centered, Z_bench sits slightly below Z_min — the small ppm on the other side pulls it down.
Z_bench 反而比 Z_min 大。数学上不可能:多加了一侧 ppm,反查回来的 Z 只会更小。Z_bench > Z_min 一定是公式或符号弄反了。
Seeing Z_bench come out larger than Z_min. Mathematically impossible: adding more tail area can only push Z down. Z_bench > Z_min always means the formula or a sign was flipped somewhere.